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\begin{document} |
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\section{Introduction} |
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Interfacial thermal conductance is extensively studied both |
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experimentally and computationally, due to its importance in nanoscale |
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science and technology. Reliability of nanoscale devices depends on |
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their thermal transport properties. Unlike bulk homogeneous materials, |
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nanoscale materials features significant presence of interfaces, and |
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these interfaces could dominate the heat transfer behavior of these |
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experimentally and computationally\cite{cahill:793}, due to its |
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importance in nanoscale science and technology. Reliability of |
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nanoscale devices depends on their thermal transport |
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properties. Unlike bulk homogeneous materials, nanoscale materials |
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features significant presence of interfaces, and these interfaces |
| 82 |
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could dominate the heat transfer behavior of these |
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|
materials. Furthermore, these materials are generally heterogeneous, |
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which challenges traditional research methods for homogeneous systems. |
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> |
which challenges traditional research methods for homogeneous |
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systems. |
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|
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Heat conductance of molecular and nano-scale interfaces will be |
| 88 |
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affected by the chemical details of the surface. Experimentally, |
| 99 |
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that specific ligands (capping agents) could completely eliminate this |
| 100 |
|
barrier ($G\rightarrow\infty$)\cite{doi:10.1021/la904855s}. |
| 101 |
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|
| 102 |
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Theoretical and computational studies were also engaged in the |
| 103 |
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interfacial thermal transport research in order to gain an |
| 104 |
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understanding of this phenomena at the molecular level. However, the |
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relatively low thermal flux through interfaces is difficult to measure |
| 106 |
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with EMD or forward NEMD simulation methods. Therefore, developing |
| 107 |
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good simulation methods will be desirable in order to investigate |
| 108 |
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thermal transport across interfaces. |
| 102 |
> |
Theoretical and computational models have also been used to study the |
| 103 |
> |
interfacial thermal transport in order to gain an understanding of |
| 104 |
> |
this phenomena at the molecular level. Recently, Hase and coworkers |
| 105 |
> |
employed Non-Equilibrium Molecular Dynamics (NEMD) simulations to |
| 106 |
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study thermal transport from hot Au(111) substrate to a self-assembled |
| 107 |
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monolayer of alkylthiol with relatively long chain (8-20 carbon |
| 108 |
> |
atoms)\cite{hase:2010,hase:2011}. However, ensemble averaged |
| 109 |
> |
measurements for heat conductance of interfaces between the capping |
| 110 |
> |
monolayer on Au and a solvent phase has yet to be studied. |
| 111 |
> |
The comparatively low thermal flux through interfaces is |
| 112 |
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difficult to measure with Equilibrium MD or forward NEMD simulation |
| 113 |
> |
methods. Therefore, the Reverse NEMD (RNEMD) methods would have the |
| 114 |
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advantage of having this difficult to measure flux known when studying |
| 115 |
> |
the thermal transport across interfaces, given that the simulation |
| 116 |
> |
methods being able to effectively apply an unphysical flux in |
| 117 |
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non-homogeneous systems. |
| 118 |
> |
|
| 119 |
|
Recently, we have developed the Non-Isotropic Velocity Scaling (NIVS) |
| 120 |
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algorithm for RNEMD simulations\cite{kuang:164101}. This algorithm |
| 121 |
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retains the desirable features of RNEMD (conservation of linear |
| 122 |
|
momentum and total energy, compatibility with periodic boundary |
| 123 |
|
conditions) while establishing true thermal distributions in each of |
| 124 |
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the two slabs. Furthermore, it allows more effective thermal exchange |
| 125 |
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between particles of different identities, and thus enables extensive |
| 126 |
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study of interfacial conductance. |
| 124 |
> |
the two slabs. Furthermore, it allows effective thermal exchange |
| 125 |
> |
between particles of different identities, and thus makes the study of |
| 126 |
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interfacial conductance much simpler. |
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|
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[BRIEF INTRO OF OUR PAPER] |
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[WHY STUDY AU-THIOL SURFACE][CITE SHAOYI JIANG] |
| 128 |
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The work presented here deals with the Au(111) surface covered to |
| 129 |
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varying degrees by butanethiol, a capping agent with short carbon |
| 130 |
> |
chain, and solvated with organic solvents of different molecular |
| 131 |
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properties. Different models were used for both the capping agent and |
| 132 |
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the solvent force field parameters. Using the NIVS algorithm, the |
| 133 |
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thermal transport across these interfaces was studied and the |
| 134 |
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underlying mechanism for this phenomena was investigated. |
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|
| 136 |
+ |
[MAY ADD WHY STUDY AU-THIOL SURFACE; CITE SHAOYI JIANG] |
| 137 |
+ |
|
| 138 |
|
\section{Methodology} |
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\subsection{Algorithm} |
| 140 |
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[BACKGROUND FOR MD METHODS] |
| 141 |
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There have been many algorithms for computing thermal conductivity |
| 142 |
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using molecular dynamics simulations. However, interfacial conductance |
| 143 |
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is at least an order of magnitude smaller. This would make the |
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calculation even more difficult for those slowly-converging |
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equilibrium methods. Imposed-flux non-equilibrium |
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\subsection{Imposd-Flux Methods in MD Simulations} |
| 140 |
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For systems with low interfacial conductivity one must have a method |
| 141 |
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capable of generating relatively small fluxes, compared to those |
| 142 |
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required for bulk conductivity. This requirement makes the calculation |
| 143 |
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even more difficult for those slowly-converging equilibrium |
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methods\cite{Viscardy:2007lq}. |
| 145 |
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Forward methods impose gradient, but in interfacail conditions it is |
| 146 |
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not clear what behavior to impose at the boundary... |
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Imposed-flux reverse non-equilibrium |
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|
methods\cite{MullerPlathe:1997xw} have the flux set {\it a priori} and |
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the response of temperature or momentum gradients are easier to |
| 150 |
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measure than the flux, if unknown, and thus, is a preferable way to |
| 151 |
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the forward NEMD methods. Although the momentum swapping approach for |
| 152 |
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flux-imposing can be used for exchanging energy between particles of |
| 153 |
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different identity, the kinetic energy transfer efficiency is affected |
| 154 |
< |
by the mass difference between the particles, which limits its |
| 134 |
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application on heterogeneous interfacial systems. |
| 149 |
> |
the thermal response becomes easier to |
| 150 |
> |
measure than the flux. Although M\"{u}ller-Plathe's original momentum |
| 151 |
> |
swapping approach can be used for exchanging energy between particles |
| 152 |
> |
of different identity, the kinetic energy transfer efficiency is |
| 153 |
> |
affected by the mass difference between the particles, which limits |
| 154 |
> |
its application on heterogeneous interfacial systems. |
| 155 |
|
|
| 156 |
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The non-isotropic velocity scaling (NIVS)\cite{kuang:164101} approach in |
| 157 |
< |
non-equilibrium MD simulations is able to impose relatively large |
| 158 |
< |
kinetic energy flux without obvious perturbation to the velocity |
| 159 |
< |
distribution of the simulated systems. Furthermore, this approach has |
| 156 |
> |
The non-isotropic velocity scaling (NIVS)\cite{kuang:164101} approach to |
| 157 |
> |
non-equilibrium MD simulations is able to impose a wide range of |
| 158 |
> |
kinetic energy fluxes without obvious perturbation to the velocity |
| 159 |
> |
distributions of the simulated systems. Furthermore, this approach has |
| 160 |
|
the advantage in heterogeneous interfaces in that kinetic energy flux |
| 161 |
|
can be applied between regions of particles of arbitary identity, and |
| 162 |
< |
the flux quantity is not restricted by particle mass difference. |
| 162 |
> |
the flux will not be restricted by difference in particle mass. |
| 163 |
|
|
| 164 |
|
The NIVS algorithm scales the velocity vectors in two separate regions |
| 165 |
|
of a simulation system with respective diagonal scaling matricies. To |
| 166 |
|
determine these scaling factors in the matricies, a set of equations |
| 167 |
|
including linear momentum conservation and kinetic energy conservation |
| 168 |
< |
constraints and target momentum/energy flux satisfaction is |
| 169 |
< |
solved. With the scaling operation applied to the system in a set |
| 170 |
< |
frequency, corresponding momentum/temperature gradients can be built, |
| 171 |
< |
which can be used for computing transportation properties and other |
| 172 |
< |
applications related to momentum/temperature gradients. The NIVS |
| 153 |
< |
algorithm conserves momenta and energy and does not depend on an |
| 154 |
< |
external thermostat. |
| 168 |
> |
constraints and target energy flux satisfaction is solved. With the |
| 169 |
> |
scaling operation applied to the system in a set frequency, bulk |
| 170 |
> |
temperature gradients can be easily established, and these can be used |
| 171 |
> |
for computing thermal conductivities. The NIVS algorithm conserves |
| 172 |
> |
momenta and energy and does not depend on an external thermostat. |
| 173 |
|
|
| 174 |
|
\subsection{Defining Interfacial Thermal Conductivity $G$} |
| 175 |
|
For interfaces with a relatively low interfacial conductance, the bulk |
| 187 |
|
T_\mathrm{cold}\rangle}$ are the average observed temperature of the |
| 188 |
|
two separated phases. |
| 189 |
|
|
| 190 |
< |
When the interfacial conductance is {\it not} small, two ways can be |
| 191 |
< |
used to define $G$. |
| 190 |
> |
When the interfacial conductance is {\it not} small, there are two |
| 191 |
> |
ways to define $G$. |
| 192 |
|
|
| 193 |
< |
One way is to assume the temperature is discretely different on two |
| 194 |
< |
sides of the interface, $G$ can be calculated with the thermal flux |
| 195 |
< |
applied $J$ and the maximum temperature difference measured along the |
| 196 |
< |
thermal gradient max($\Delta T$), which occurs at the interface, as: |
| 193 |
> |
One way is to assume the temperature is discrete on the two sides of |
| 194 |
> |
the interface. $G$ can be calculated using the applied thermal flux |
| 195 |
> |
$J$ and the maximum temperature difference measured along the thermal |
| 196 |
> |
gradient max($\Delta T$), which occurs at the Gibbs deviding surface, |
| 197 |
> |
as: |
| 198 |
|
\begin{equation} |
| 199 |
|
G=\frac{J}{\Delta T} |
| 200 |
|
\label{discreteG} |
| 215 |
|
|
| 216 |
|
With the temperature profile obtained from simulations, one is able to |
| 217 |
|
approximate the first and second derivatives of $T$ with finite |
| 218 |
< |
difference method and thus calculate $G^\prime$. |
| 218 |
> |
difference methods and thus calculate $G^\prime$. |
| 219 |
|
|
| 220 |
< |
In what follows, both definitions are used for calculation and comparison. |
| 220 |
> |
In what follows, both definitions have been used for calculation and |
| 221 |
> |
are compared in the results. |
| 222 |
|
|
| 223 |
< |
[IMPOSE G DEFINITION INTO OUR SYSTEMS] |
| 224 |
< |
To facilitate the use of the above definitions in calculating $G$ and |
| 225 |
< |
$G^\prime$, we have a metal slab with its (111) surfaces perpendicular |
| 226 |
< |
to the $z$-axis of our simulation cells. With or withour capping |
| 227 |
< |
agents on the surfaces, the metal slab is solvated with organic |
| 208 |
< |
solvents, as illustrated in Figure \ref{demoPic}. |
| 223 |
> |
To compare the above definitions ($G$ and $G^\prime$), we have modeled |
| 224 |
> |
a metal slab with its (111) surfaces perpendicular to the $z$-axis of |
| 225 |
> |
our simulation cells. Both with and withour capping agents on the |
| 226 |
> |
surfaces, the metal slab is solvated with simple organic solvents, as |
| 227 |
> |
illustrated in Figure \ref{demoPic}. |
| 228 |
|
|
| 229 |
|
\begin{figure} |
| 230 |
< |
\includegraphics[width=\linewidth]{demoPic} |
| 231 |
< |
\caption{A sample showing how a metal slab has its (111) surface |
| 232 |
< |
covered by capping agent molecules and solvated by hexane.} |
| 230 |
> |
\includegraphics[width=\linewidth]{method} |
| 231 |
> |
\caption{Interfacial conductance can be calculated by applying an |
| 232 |
> |
(unphysical) kinetic energy flux between two slabs, one located |
| 233 |
> |
within the metal and another on the edge of the periodic box. The |
| 234 |
> |
system responds by forming a thermal response or a gradient. In |
| 235 |
> |
bulk liquids, this gradient typically has a single slope, but in |
| 236 |
> |
interfacial systems, there are distinct thermal conductivity |
| 237 |
> |
domains. The interfacial conductance, $G$ is found by measuring the |
| 238 |
> |
temperature gap at the Gibbs dividing surface, or by using second |
| 239 |
> |
derivatives of the thermal profile.} |
| 240 |
|
\label{demoPic} |
| 241 |
|
\end{figure} |
| 242 |
|
|
| 243 |
< |
With a simulation cell setup following the above manner, one is able |
| 244 |
< |
to equilibrate the system and impose an unphysical thermal flux |
| 245 |
< |
between the liquid and the metal phase with the NIVS algorithm. Under |
| 246 |
< |
a stablized thermal gradient induced by periodically applying the |
| 247 |
< |
unphysical flux, one is able to obtain a temperature profile and the |
| 248 |
< |
physical thermal flux corresponding to it, which equals to the |
| 249 |
< |
unphysical flux applied by NIVS. These data enables the evaluation of |
| 250 |
< |
the interfacial thermal conductance of a surface. Figure \ref{gradT} |
| 225 |
< |
is an example how those stablized thermal gradient can be used to |
| 226 |
< |
obtain the 1st and 2nd derivatives of the temperature profile. |
| 243 |
> |
With the simulation cell described above, we are able to equilibrate |
| 244 |
> |
the system and impose an unphysical thermal flux between the liquid |
| 245 |
> |
and the metal phase using the NIVS algorithm. By periodically applying |
| 246 |
> |
the unphysical flux, we are able to obtain a temperature profile and |
| 247 |
> |
its spatial derivatives. These quantities enable the evaluation of the |
| 248 |
> |
interfacial thermal conductance of a surface. Figure \ref{gradT} is an |
| 249 |
> |
example how those applied thermal fluxes can be used to obtain the 1st |
| 250 |
> |
and 2nd derivatives of the temperature profile. |
| 251 |
|
|
| 252 |
|
\begin{figure} |
| 253 |
|
\includegraphics[width=\linewidth]{gradT} |
| 256 |
|
\label{gradT} |
| 257 |
|
\end{figure} |
| 258 |
|
|
| 235 |
– |
[MAY INCLUDE POWER SPECTRUM PROTOCOL] |
| 236 |
– |
|
| 259 |
|
\section{Computational Details} |
| 260 |
|
\subsection{Simulation Protocol} |
| 261 |
< |
In our simulations, Au is used to construct a metal slab with bare |
| 262 |
< |
(111) surface perpendicular to the $z$-axis. Different slab thickness |
| 263 |
< |
(layer numbers of Au) are simulated. This metal slab is first |
| 264 |
< |
equilibrated under normal pressure (1 atm) and a desired |
| 265 |
< |
temperature. After equilibration, butanethiol is used as the capping |
| 266 |
< |
agent molecule to cover the bare Au (111) surfaces evenly. The sulfur |
| 267 |
< |
atoms in the butanethiol molecules would occupy the three-fold sites |
| 268 |
< |
of the surfaces, and the maximal butanethiol capacity on Au surface is |
| 269 |
< |
$1/3$ of the total number of surface Au atoms[CITATION]. A series of |
| 270 |
< |
different coverage surfaces is investigated in order to study the |
| 271 |
< |
relation between coverage and conductance. |
| 261 |
> |
The NIVS algorithm has been implemented in our MD simulation code, |
| 262 |
> |
OpenMD\cite{Meineke:2005gd,openmd}, and was used for our |
| 263 |
> |
simulations. Different slab thickness (layer numbers of Au) were |
| 264 |
> |
simulated. Metal slabs were first equilibrated under atmospheric |
| 265 |
> |
pressure (1 atm) and a desired temperature (e.g. 200K). After |
| 266 |
> |
equilibration, butanethiol capping agents were placed at three-fold |
| 267 |
> |
sites on the Au(111) surfaces. The maximum butanethiol capacity on Au |
| 268 |
> |
surface is $1/3$ of the total number of surface Au |
| 269 |
> |
atoms\cite{vlugt:cpc2007154}. A series of different coverages was |
| 270 |
> |
investigated in order to study the relation between coverage and |
| 271 |
> |
interfacial conductance. |
| 272 |
|
|
| 273 |
< |
[COVERAGE DISCRIPTION] However, since the interactions between surface |
| 274 |
< |
Au and butanethiol is non-bonded, the capping agent molecules are |
| 275 |
< |
allowed to migrate to an empty neighbor three-fold site during a |
| 276 |
< |
simulation. Therefore, the initial configuration would not severely |
| 277 |
< |
affect the sampling of a variety of configurations of the same |
| 278 |
< |
coverage, and the final conductance measurement would be an average |
| 279 |
< |
effect of these configurations explored in the simulations. [MAY NEED FIGURES] |
| 273 |
> |
The capping agent molecules were allowed to migrate during the |
| 274 |
> |
simulations. They distributed themselves uniformly and sampled a |
| 275 |
> |
number of three-fold sites throughout out study. Therefore, the |
| 276 |
> |
initial configuration would not noticeably affect the sampling of a |
| 277 |
> |
variety of configurations of the same coverage, and the final |
| 278 |
> |
conductance measurement would be an average effect of these |
| 279 |
> |
configurations explored in the simulations. [MAY NEED FIGURES] |
| 280 |
|
|
| 281 |
< |
After the modified Au-butanethiol surface systems are equilibrated |
| 282 |
< |
under canonical ensemble, Packmol\cite{packmol} is used to pack |
| 283 |
< |
organic solvent molecules in the previously vacuum part of the |
| 284 |
< |
simulation cells, which guarantees that short range repulsive |
| 285 |
< |
interactions do not disrupt the simulations. Two solvents are |
| 286 |
< |
investigated, one which has little vibrational overlap with the |
| 265 |
< |
alkanethiol and plane-like shape (toluene), and one which has similar |
| 266 |
< |
vibrational frequencies and chain-like shape ({\it n}-hexane). [MAY |
| 267 |
< |
EXPLAIN WHY WE CHOOSE THEM] |
| 281 |
> |
After the modified Au-butanethiol surface systems were equilibrated |
| 282 |
> |
under canonical ensemble, organic solvent molecules were packed in the |
| 283 |
> |
previously empty part of the simulation cells\cite{packmol}. Two |
| 284 |
> |
solvents were investigated, one which has little vibrational overlap |
| 285 |
> |
with the alkanethiol and a planar shape (toluene), and one which has |
| 286 |
> |
similar vibrational frequencies and chain-like shape ({\it n}-hexane). |
| 287 |
|
|
| 288 |
< |
The spacing filled by solvent molecules, i.e. the gap between |
| 288 |
> |
The space filled by solvent molecules, i.e. the gap between |
| 289 |
|
periodically repeated Au-butanethiol surfaces should be carefully |
| 290 |
|
chosen. A very long length scale for the thermal gradient axis ($z$) |
| 291 |
|
may cause excessively hot or cold temperatures in the middle of the |
| 330 |
|
same type of particles and between particles of different species. |
| 331 |
|
|
| 332 |
|
The Au-Au interactions in metal lattice slab is described by the |
| 333 |
< |
quantum Sutton-Chen (QSC) formulation.\cite{PhysRevB.59.3527} The QSC |
| 333 |
> |
quantum Sutton-Chen (QSC) formulation\cite{PhysRevB.59.3527}. The QSC |
| 334 |
|
potentials include zero-point quantum corrections and are |
| 335 |
|
reparametrized for accurate surface energies compared to the |
| 336 |
|
Sutton-Chen potentials\cite{Chen90}. |
| 337 |
|
|
| 338 |
< |
Figure [REF] demonstrates how we name our pseudo-atoms of the |
| 339 |
< |
molecules in our simulations. |
| 321 |
< |
[FIGURE FOR MOLECULE NOMENCLATURE] |
| 338 |
> |
Figure \ref{demoMol} demonstrates how we name our pseudo-atoms of the |
| 339 |
> |
organic solvent molecules in our simulations. |
| 340 |
|
|
| 341 |
+ |
\begin{figure} |
| 342 |
+ |
\includegraphics[width=\linewidth]{structures} |
| 343 |
+ |
\caption{Structures of the capping agent and solvents utilized in |
| 344 |
+ |
these simulations. The chemically-distinct sites (a-e) are expanded |
| 345 |
+ |
in terms of constituent atoms for both United Atom (UA) and All Atom |
| 346 |
+ |
(AA) force fields. Most parameters are from |
| 347 |
+ |
Refs. \protect\cite{TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes} (UA) and |
| 348 |
+ |
\protect\cite{OPLSAA} (AA). Cross-interactions with the Au atoms are given |
| 349 |
+ |
in Table \ref{MnM}.} |
| 350 |
+ |
\label{demoMol} |
| 351 |
+ |
\end{figure} |
| 352 |
+ |
|
| 353 |
|
For both solvent molecules, straight chain {\it n}-hexane and aromatic |
| 354 |
|
toluene, United-Atom (UA) and All-Atom (AA) models are used |
| 355 |
|
respectively. The TraPPE-UA |
| 381 |
|
parameters for alkyl thiols. However, alkyl thiols adsorbed on Au(111) |
| 382 |
|
surfaces do not have the hydrogen atom bonded to sulfur. To adapt this |
| 383 |
|
change and derive suitable parameters for butanethiol adsorbed on |
| 384 |
< |
Au(111) surfaces, we adopt the S parameters from [CITATION CF VLUGT] |
| 385 |
< |
and modify parameters for its neighbor C atom for charge balance in |
| 386 |
< |
the molecule. Note that the model choice (UA or AA) of capping agent |
| 387 |
< |
can be different from the solvent. Regardless of model choice, the |
| 388 |
< |
force field parameters for interactions between capping agent and |
| 389 |
< |
solvent can be derived using Lorentz-Berthelot Mixing Rule: |
| 384 |
> |
Au(111) surfaces, we adopt the S parameters from Luedtke and |
| 385 |
> |
Landman\cite{landman:1998} and modify parameters for its neighbor C |
| 386 |
> |
atom for charge balance in the molecule. Note that the model choice |
| 387 |
> |
(UA or AA) of capping agent can be different from the |
| 388 |
> |
solvent. Regardless of model choice, the force field parameters for |
| 389 |
> |
interactions between capping agent and solvent can be derived using |
| 390 |
> |
Lorentz-Berthelot Mixing Rule: |
| 391 |
> |
\begin{eqnarray} |
| 392 |
> |
\sigma_{ij} & = & \frac{1}{2} \left(\sigma_{ii} + \sigma_{jj}\right) \\ |
| 393 |
> |
\epsilon_{ij} & = & \sqrt{\epsilon_{ii}\epsilon_{jj}} |
| 394 |
> |
\end{eqnarray} |
| 395 |
|
|
| 361 |
– |
|
| 396 |
|
To describe the interactions between metal Au and non-metal capping |
| 397 |
|
agent and solvent particles, we refer to an adsorption study of alkyl |
| 398 |
|
thiols on gold surfaces by Vlugt {\it et |
| 399 |
|
al.}\cite{vlugt:cpc2007154} They fitted an effective Lennard-Jones |
| 400 |
|
form of potential parameters for the interaction between Au and |
| 401 |
|
pseudo-atoms CH$_x$ and S based on a well-established and widely-used |
| 402 |
< |
effective potential of Hautman and Klein[CITATION] for the Au(111) |
| 403 |
< |
surface. As our simulations require the gold lattice slab to be |
| 404 |
< |
non-rigid so that it could accommodate kinetic energy for thermal |
| 402 |
> |
effective potential of Hautman and Klein\cite{hautman:4994} for the |
| 403 |
> |
Au(111) surface. As our simulations require the gold lattice slab to |
| 404 |
> |
be non-rigid so that it could accommodate kinetic energy for thermal |
| 405 |
|
transport study purpose, the pair-wise form of potentials is |
| 406 |
|
preferred. |
| 407 |
|
|
| 420 |
|
\begin{table*} |
| 421 |
|
\begin{minipage}{\linewidth} |
| 422 |
|
\begin{center} |
| 423 |
< |
\caption{Lennard-Jones parameters for Au-non-Metal |
| 424 |
< |
interactions in our simulations.} |
| 425 |
< |
|
| 426 |
< |
\begin{tabular}{ccc} |
| 423 |
> |
\caption{Non-bonded interaction parameters (including cross |
| 424 |
> |
interactions with Au atoms) for both force fields used in this |
| 425 |
> |
work.} |
| 426 |
> |
\begin{tabular}{lllllll} |
| 427 |
|
\hline\hline |
| 428 |
< |
Non-metal atom & $\sigma$ & $\epsilon$ \\ |
| 429 |
< |
(or pseudo-atom) & \AA & kcal/mol \\ |
| 428 |
> |
& Site & $\sigma_{ii}$ & $\epsilon_{ii}$ & $q_i$ & |
| 429 |
> |
$\sigma_{Au-i}$ & $\epsilon_{Au-i}$ \\ |
| 430 |
> |
& & (\AA) & (kcal/mol) & ($e$) & (\AA) & (kcal/mol) \\ |
| 431 |
|
\hline |
| 432 |
< |
S & 2.40 & 8.465 \\ |
| 433 |
< |
CH3 & 3.54 & 0.2146 \\ |
| 434 |
< |
CH2 & 3.54 & 0.1749 \\ |
| 435 |
< |
CT3 & 3.365 & 0.1373 \\ |
| 436 |
< |
CT2 & 3.365 & 0.1373 \\ |
| 437 |
< |
CTT & 3.365 & 0.1373 \\ |
| 438 |
< |
HC & 2.865 & 0.09256 \\ |
| 439 |
< |
CHar & 3.4625 & 0.1680 \\ |
| 440 |
< |
CRar & 3.555 & 0.1604 \\ |
| 441 |
< |
CA & 3.173 & 0.0640 \\ |
| 442 |
< |
HA & 2.746 & 0.0414 \\ |
| 432 |
> |
United Atom (UA) |
| 433 |
> |
&CH3 & 3.75 & 0.1947 & - & 3.54 & 0.2146 \\ |
| 434 |
> |
&CH2 & 3.95 & 0.0914 & - & 3.54 & 0.1749 \\ |
| 435 |
> |
&CHar & 3.695 & 0.1003 & - & 3.4625 & 0.1680 \\ |
| 436 |
> |
&CRar & 3.88 & 0.04173 & - & 3.555 & 0.1604 \\ |
| 437 |
> |
\hline |
| 438 |
> |
All Atom (AA) |
| 439 |
> |
&CT3 & 3.50 & 0.066 & -0.18 & 3.365 & 0.1373 \\ |
| 440 |
> |
&CT2 & 3.50 & 0.066 & -0.12 & 3.365 & 0.1373 \\ |
| 441 |
> |
&CTT & 3.50 & 0.066 & -0.065 & 3.365 & 0.1373 \\ |
| 442 |
> |
&HC & 2.50 & 0.030 & 0.06 & 2.865 & 0.09256 \\ |
| 443 |
> |
&CA & 3.55 & 0.070 & -0.115 & 3.173 & 0.0640 \\ |
| 444 |
> |
&HA & 2.42 & 0.030 & 0.115 & 2.746 & 0.0414 \\ |
| 445 |
> |
\hline |
| 446 |
> |
Both UA and AA & S & 4.45 & 0.25 & - & 2.40 & 8.465 \\ |
| 447 |
|
\hline\hline |
| 448 |
|
\end{tabular} |
| 449 |
|
\label{MnM} |
| 513 |
|
interfaces with UA model and different hexane molecule numbers |
| 514 |
|
at different temperatures using a range of energy fluxes.} |
| 515 |
|
|
| 516 |
< |
\begin{tabular}{cccccccc} |
| 516 |
> |
\begin{tabular}{ccccccc} |
| 517 |
|
\hline\hline |
| 518 |
< |
$\langle T\rangle$ & & $L_x$ & $L_y$ & $L_z$ & $J_z$ & |
| 519 |
< |
$G$ & $G^\prime$ \\ |
| 520 |
< |
(K) & $N_{hexane}$ & \multicolumn{3}{c}{(\AA)} & (GW/m$^2$) & |
| 518 |
> |
$\langle T\rangle$ & $N_{hexane}$ & Fixed & $\rho_{hexane}$ & |
| 519 |
> |
$J_z$ & $G$ & $G^\prime$ \\ |
| 520 |
> |
(K) & & $L_x$ \& $L_y$? & (g/cm$^3$) & (GW/m$^2$) & |
| 521 |
|
\multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
| 522 |
|
\hline |
| 523 |
< |
200 & 266 & 29.86 & 25.80 & 113.1 & -0.96 & |
| 524 |
< |
102() & 80.0() \\ |
| 525 |
< |
& 200 & 29.84 & 25.81 & 93.9 & 1.92 & |
| 526 |
< |
129() & 87.3() \\ |
| 527 |
< |
& & 29.84 & 25.81 & 95.3 & 1.93 & |
| 528 |
< |
131() & 77.5() \\ |
| 529 |
< |
& 166 & 29.84 & 25.81 & 85.7 & 0.97 & |
| 530 |
< |
115() & 69.3() \\ |
| 531 |
< |
& & & & & 1.94 & |
| 532 |
< |
125() & 87.1() \\ |
| 533 |
< |
250 & 200 & 29.84 & 25.87 & 106.8 & 0.96 & |
| 534 |
< |
81.8() & 67.0() \\ |
| 535 |
< |
& 166 & 29.87 & 25.84 & 94.8 & 0.98 & |
| 536 |
< |
79.0() & 62.9() \\ |
| 537 |
< |
& & 29.84 & 25.85 & 95.0 & 1.44 & |
| 538 |
< |
76.2() & 64.8() \\ |
| 523 |
> |
200 & 266 & No & 0.672 & -0.96 & 102() & 80.0() \\ |
| 524 |
> |
& 200 & Yes & 0.694 & 1.92 & 129() & 87.3() \\ |
| 525 |
> |
& & Yes & 0.672 & 1.93 & 131() & 77.5() \\ |
| 526 |
> |
& & No & 0.688 & 0.96 & 125() & 90.2() \\ |
| 527 |
> |
& & & & 1.91 & 139() & 101() \\ |
| 528 |
> |
& & & & 2.83 & 141() & 89.9() \\ |
| 529 |
> |
& 166 & Yes & 0.679 & 0.97 & 115() & 69.3() \\ |
| 530 |
> |
& & & & 1.94 & 125() & 87.1() \\ |
| 531 |
> |
& & No & 0.681 & 0.97 & 141() & 77.7() \\ |
| 532 |
> |
& & & & 1.92 & 138() & 98.9() \\ |
| 533 |
> |
\hline |
| 534 |
> |
250 & 200 & No & 0.560 & 0.96 & 74.8() & 61.8() \\ |
| 535 |
> |
& & & & -0.95 & 49.4() & 45.7() \\ |
| 536 |
> |
& 166 & Yes & 0.570 & 0.98 & 79.0() & 62.9() \\ |
| 537 |
> |
& & No & 0.569 & 0.97 & 80.3() & 67.1() \\ |
| 538 |
> |
& & & & 1.44 & 76.2() & 64.8() \\ |
| 539 |
> |
& & & & -0.95 & 56.4() & 54.4() \\ |
| 540 |
> |
& & & & -1.85 & 47.8() & 53.5() \\ |
| 541 |
|
\hline\hline |
| 542 |
|
\end{tabular} |
| 543 |
|
\label{AuThiolHexaneUA} |
| 568 |
|
important role in the thermal transport process across the interface |
| 569 |
|
in that higher degree of contact could yield increased conductance. |
| 570 |
|
|
| 571 |
< |
[ADD Lxyz AND ERROR ESTIMATE TO TABLE] |
| 571 |
> |
[ADD ERROR ESTIMATE TO TABLE] |
| 572 |
|
\begin{table*} |
| 573 |
|
\begin{minipage}{\linewidth} |
| 574 |
|
\begin{center} |
| 577 |
|
interface at different temperatures using a range of energy |
| 578 |
|
fluxes.} |
| 579 |
|
|
| 580 |
< |
\begin{tabular}{cccc} |
| 580 |
> |
\begin{tabular}{ccccc} |
| 581 |
|
\hline\hline |
| 582 |
< |
$\langle T\rangle$ & $J_z$ & $G$ & $G^\prime$ \\ |
| 583 |
< |
(K) & (GW/m$^2$) & \multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
| 582 |
> |
$\langle T\rangle$ & $\rho_{toluene}$ & $J_z$ & $G$ & $G^\prime$ \\ |
| 583 |
> |
(K) & (g/cm$^3$) & (GW/m$^2$) & \multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
| 584 |
|
\hline |
| 585 |
< |
200 & -1.86 & 180() & 135() \\ |
| 586 |
< |
& 2.15 & 204() & 113() \\ |
| 587 |
< |
& -3.93 & 175() & 114() \\ |
| 588 |
< |
300 & -1.91 & 143() & 125() \\ |
| 589 |
< |
& -4.19 & 134() & 113() \\ |
| 585 |
> |
200 & 0.933 & -1.86 & 180() & 135() \\ |
| 586 |
> |
& & 2.15 & 204() & 113() \\ |
| 587 |
> |
& & -3.93 & 175() & 114() \\ |
| 588 |
> |
\hline |
| 589 |
> |
300 & 0.855 & -1.91 & 143() & 125() \\ |
| 590 |
> |
& & -4.19 & 134() & 113() \\ |
| 591 |
|
\hline\hline |
| 592 |
|
\end{tabular} |
| 593 |
|
\label{AuThiolToluene} |
| 620 |
|
However, when the surface is not completely covered by butanethiols, |
| 621 |
|
the simulated system is more resistent to the reconstruction |
| 622 |
|
above. Our Au-butanethiol/toluene system did not see this phenomena |
| 623 |
< |
even at $<T>\sim$300K. The Au(111) surfaces have a 90\% coverage of |
| 624 |
< |
butanethiols and have empty three-fold sites. These empty sites could |
| 625 |
< |
help prevent surface reconstruction in that they provide other means |
| 626 |
< |
of capping agent relaxation. It is observed that butanethiols can |
| 627 |
< |
migrate to their neighbor empty sites during a simulation. Therefore, |
| 628 |
< |
we were able to obtain $G$'s for these interfaces even at a relatively |
| 629 |
< |
high temperature without being affected by surface reconstructions. |
| 623 |
> |
even at $\langle T\rangle\sim$300K. The Au(111) surfaces have a 90\% |
| 624 |
> |
coverage of butanethiols and have empty three-fold sites. These empty |
| 625 |
> |
sites could help prevent surface reconstruction in that they provide |
| 626 |
> |
other means of capping agent relaxation. It is observed that |
| 627 |
> |
butanethiols can migrate to their neighbor empty sites during a |
| 628 |
> |
simulation. Therefore, we were able to obtain $G$'s for these |
| 629 |
> |
interfaces even at a relatively high temperature without being |
| 630 |
> |
affected by surface reconstructions. |
| 631 |
|
|
| 632 |
|
\subsection{Influence of Capping Agent Coverage on $G$} |
| 633 |
|
To investigate the influence of butanethiol coverage on interfacial |
| 703 |
|
its effect to the process of interfacial thermal transport. Thus, one |
| 704 |
|
can see a plateau of $G$ vs. butanethiol coverage in our results. |
| 705 |
|
|
| 706 |
< |
[NEED ERROR ESTIMATE, MAY ALSO PUT J HERE] |
| 707 |
< |
\begin{table*} |
| 708 |
< |
\begin{minipage}{\linewidth} |
| 709 |
< |
\begin{center} |
| 710 |
< |
\caption{Computed interfacial thermal conductivity ($G$) values |
| 711 |
< |
for the Au-butanethiol/solvent interface with various UA |
| 712 |
< |
models and different capping agent coverages at $\langle |
| 713 |
< |
T\rangle\sim$200K using certain energy flux respectively.} |
| 671 |
< |
|
| 672 |
< |
\begin{tabular}{cccc} |
| 673 |
< |
\hline\hline |
| 674 |
< |
Thiol & \multicolumn{3}{c}{$G$(MW/m$^2$/K)} \\ |
| 675 |
< |
coverage (\%) & hexane & hexane(D) & toluene \\ |
| 676 |
< |
\hline |
| 677 |
< |
0.0 & 46.5() & 43.9() & 70.1() \\ |
| 678 |
< |
25.0 & 151() & 153() & 249() \\ |
| 679 |
< |
50.0 & 172() & 182() & 214() \\ |
| 680 |
< |
75.0 & 242() & 229() & 244() \\ |
| 681 |
< |
88.9 & 178() & - & - \\ |
| 682 |
< |
100.0 & 137() & 153() & 187() \\ |
| 683 |
< |
\hline\hline |
| 684 |
< |
\end{tabular} |
| 685 |
< |
\label{tlnUhxnUhxnD} |
| 686 |
< |
\end{center} |
| 687 |
< |
\end{minipage} |
| 688 |
< |
\end{table*} |
| 706 |
> |
\begin{figure} |
| 707 |
> |
\includegraphics[width=\linewidth]{coverage} |
| 708 |
> |
\caption{Comparison of interfacial thermal conductivity ($G$) values |
| 709 |
> |
for the Au-butanethiol/solvent interface with various UA models and |
| 710 |
> |
different capping agent coverages at $\langle T\rangle\sim$200K |
| 711 |
> |
using certain energy flux respectively.} |
| 712 |
> |
\label{coverage} |
| 713 |
> |
\end{figure} |
| 714 |
|
|
| 715 |
|
\subsection{Influence of Chosen Molecule Model on $G$} |
| 716 |
|
[MAY COMBINE W MECHANISM STUDY] |
| 723 |
|
the previous section. Table \ref{modelTest} summarizes the results of |
| 724 |
|
these studies. |
| 725 |
|
|
| 701 |
– |
[MORE DATA; ERROR ESTIMATE] |
| 726 |
|
\begin{table*} |
| 727 |
|
\begin{minipage}{\linewidth} |
| 728 |
|
\begin{center} |
| 730 |
|
\caption{Computed interfacial thermal conductivity ($G$ and |
| 731 |
|
$G^\prime$) values for interfaces using various models for |
| 732 |
|
solvent and capping agent (or without capping agent) at |
| 733 |
< |
$\langle T\rangle\sim$200K.} |
| 733 |
> |
$\langle T\rangle\sim$200K. (D stands for deuterated solvent |
| 734 |
> |
or capping agent molecules; ``Avg.'' denotes results that are |
| 735 |
> |
averages of simulations under different $J_z$'s. Error |
| 736 |
> |
estimates indicated in parenthesis.)} |
| 737 |
|
|
| 738 |
< |
\begin{tabular}{ccccc} |
| 738 |
> |
\begin{tabular}{llccc} |
| 739 |
|
\hline\hline |
| 740 |
|
Butanethiol model & Solvent & $J_z$ & $G$ & $G^\prime$ \\ |
| 741 |
|
(or bare surface) & model & (GW/m$^2$) & |
| 742 |
|
\multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
| 743 |
|
\hline |
| 744 |
< |
UA & AA hexane & 1.94 & 135() & 129() \\ |
| 745 |
< |
& & 2.86 & 126() & 115() \\ |
| 746 |
< |
& AA toluene & 1.89 & 200() & 149() \\ |
| 747 |
< |
AA & UA hexane & 1.94 & 116() & 129() \\ |
| 748 |
< |
& AA hexane & 3.76 & 451() & 378() \\ |
| 749 |
< |
& & 4.71 & 432() & 334() \\ |
| 750 |
< |
& AA toluene & 3.79 & 487() & 290() \\ |
| 751 |
< |
AA(D) & UA hexane & 1.94 & 158() & 172() \\ |
| 752 |
< |
bare & AA hexane & 0.96 & 31.0() & 29.4() \\ |
| 744 |
> |
UA & UA hexane & Avg. & 131(9) & 87(10) \\ |
| 745 |
> |
& UA hexane(D) & 1.95 & 153(5) & 136(13) \\ |
| 746 |
> |
& AA hexane & Avg. & 131(6) & 122(10) \\ |
| 747 |
> |
& UA toluene & 1.96 & 187(16) & 151(11) \\ |
| 748 |
> |
& AA toluene & 1.89 & 200(36) & 149(53) \\ |
| 749 |
> |
\hline |
| 750 |
> |
AA & UA hexane & 1.94 & 116(9) & 129(8) \\ |
| 751 |
> |
& AA hexane & Avg. & 442(14) & 356(31) \\ |
| 752 |
> |
& AA hexane(D) & 1.93 & 222(12) & 234(54) \\ |
| 753 |
> |
& UA toluene & 1.98 & 125(25) & 97(60) \\ |
| 754 |
> |
& AA toluene & 3.79 & 487(56) & 290(42) \\ |
| 755 |
> |
\hline |
| 756 |
> |
AA(D) & UA hexane & 1.94 & 158(25) & 172(4) \\ |
| 757 |
> |
& AA hexane & 1.92 & 243(29) & 191(11) \\ |
| 758 |
> |
& AA toluene & 1.93 & 364(36) & 322(67) \\ |
| 759 |
> |
\hline |
| 760 |
> |
bare & UA hexane & Avg. & 46.5(3.2) & 49.4(4.5) \\ |
| 761 |
> |
& UA hexane(D) & 0.98 & 43.9(4.6) & 43.0(2.0) \\ |
| 762 |
> |
& AA hexane & 0.96 & 31.0(1.4) & 29.4(1.3) \\ |
| 763 |
> |
& UA toluene & 1.99 & 70.1(1.3) & 65.8(0.5) \\ |
| 764 |
|
\hline\hline |
| 765 |
|
\end{tabular} |
| 766 |
|
\label{modelTest} |
| 797 |
|
|
| 798 |
|
However, for Au-butanethiol/toluene interfaces, having the AA |
| 799 |
|
butanethiol deuterated did not yield a significant change in the |
| 800 |
< |
measurement results. |
| 801 |
< |
. , so extra degrees of freedom |
| 802 |
< |
such as the C-H vibrations could enhance heat exchange between these |
| 803 |
< |
two phases and result in a much higher conductivity. |
| 800 |
> |
measurement results. Compared to the C-H vibrational overlap between |
| 801 |
> |
hexane and butanethiol, both of which have alkyl chains, that overlap |
| 802 |
> |
between toluene and butanethiol is not so significant and thus does |
| 803 |
> |
not have as much contribution to the ``Intramolecular Vibration |
| 804 |
> |
Redistribution''[CITE HASE]. Conversely, extra degrees of freedom such |
| 805 |
> |
as the C-H vibrations could yield higher heat exchange rate between |
| 806 |
> |
these two phases and result in a much higher conductivity. |
| 807 |
|
|
| 767 |
– |
|
| 808 |
|
Although the QSC model for Au is known to predict an overly low value |
| 809 |
< |
for bulk metal gold conductivity[CITE NIVSRNEMD], our computational |
| 809 |
> |
for bulk metal gold conductivity\cite{kuang:164101}, our computational |
| 810 |
|
results for $G$ and $G^\prime$ do not seem to be affected by this |
| 811 |
< |
drawback of the model for metal. Instead, the modeling of interfacial |
| 812 |
< |
thermal transport behavior relies mainly on an accurate description of |
| 813 |
< |
the interactions between components occupying the interfaces. |
| 811 |
> |
drawback of the model for metal. Instead, our results suggest that the |
| 812 |
> |
modeling of interfacial thermal transport behavior relies mainly on |
| 813 |
> |
the accuracy of the interaction descriptions between components |
| 814 |
> |
occupying the interfaces. |
| 815 |
|
|
| 816 |
|
\subsection{Mechanism of Interfacial Thermal Conductance Enhancement |
| 817 |
|
by Capping Agent} |
| 827 |
|
the velocity auto-correlation functions, which is used to construct a |
| 828 |
|
power spectrum via a Fourier transform. |
| 829 |
|
|
| 830 |
+ |
[MAY RELATE TO HASE'S] |
| 831 |
|
The gold surfaces covered by |
| 832 |
|
butanethiol molecules, compared to bare gold surfaces, exhibit an |
| 833 |
|
additional peak observed at a frequency of $\sim$170cm$^{-1}$, which |
| 840 |
|
combination of these two effects produces the drastic interfacial |
| 841 |
|
thermal conductance enhancement in the all-atom model. |
| 842 |
|
|
| 843 |
< |
[MAY NEED TO CONVERT TO JPEG] |
| 843 |
> |
[REDO. MAY NEED TO CONVERT TO JPEG] |
| 844 |
|
\begin{figure} |
| 845 |
|
\includegraphics[width=\linewidth]{vibration} |
| 846 |
|
\caption{Vibrational spectra obtained for gold in different |
